# Professor of Applied Math Jinqiao Duan Publishes Article in Nature Communications

Jinqiao Duan, professor of applied mathematics at Illinois Institute of Technology, co-authored an article, “Restoration of Rhythmicity in Diffusively Coupled Dynamical Networks,” which was recently published in Nature Communications. Collaborators on the article are a team of international scientists from China, Germany, India, Macedonia, Russia, the United Kingdom, and the United States.

The article details the interaction of coupled oscillating systems. Oscillatory movements or rhythms are essential for various systems. When a rhythm stalls, the effect can be fatal. In a power grid it can mean a blackout, and in the human heart even death. Duan’s research team has developed a new approach for revoking these undesired effects. They use dynamical systems tools, combined with analysis and simulation, and demonstrate the new approach in experiments with chemical reactions.

• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.
• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.
• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.
• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.
• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.
• (a-f) The bifurcation diagrams of the steady-state solutions for α=1, 0.999, 0.998, 0.997, 0.996 and 0.995, respectively. The bold red lines represent stable IHSS (OD), and the thin black lines denote unstable steady states. Even an infinitesimal change of α from unity drastically shrinks the stable OD interval, which shows the high efficiency of the method in restoring rhythmic activity. The frequency is used as w=10.

On the article’s publication date, Physics.org posted an article, “From Power Grids to Heartbeat: Using Mathematics to Restore Rhythm,” reviewing the research project. In the review, Wei Zou from Huazhong University, lead author of the study, says, “I have been surprised how simple and robust our method is. Now we hope it will open a door for future research in the field of complex systems science, and eventually invoke applications in many areas ranging from biology via engineering to social sciences.”